Entire nodal solutions of a semilinear elliptic equation and their effect on concentration phenomena

We exhibit a new concentration phenomenon for the problem−ε2Δu+u=|u|p−2u in Ω, u=0 on ∂Ω,−ε2Δu+u=|u|p−2u in Ω, u=0 on ∂Ω, where Ω is some bounded smooth domain in RNRN, N≥4N≥4, ε>0ε>0, and p∈(2,2NN−2). We show that there is a sequence of sign changing solutions, which concentrate at a single point, whose asymptotic profile as ε→0ε→0 is a rescaling of a nonradial sign changing solution to the limit problem−Δu+u=|u|p−2u, u∈H1(RN),−Δu+u=|u|p−2u, u∈H1(RN), with sufficiently small energy.